Generating sets for ideals of finite type in \(H^\infty\)
In the algebra \(H^\infty\) of bounded analytic functions in the unit disc \(\mathbb D\), let \(I(f_1,\dots,f_n)\) denote the ideal generated by the elements \(f_1,\dots,f_n\), and let \(J(f_1,\dots,f_n)\) denote the a priori larger ideal NEWLINE\[NEWLINE \left\{ f\in H^\infty: \exists C>0 \text{ such that } |f|\leq C\sum_{j=1}^n |f_j| \right\} NEWLINE\]NEWLINE -- a so-called ideal of finite type.NEWLINENEWLINEThis well-written paper is mainly concerned with the conditions under which \(I=J\). The author previously [Bull. Aust. Math. Soc. 50, No. 3, 521--528 (1994; Zbl 0823.46051)] settled the corresponding question in the disc algebra. There is also previous work on the question by von Renteln [\textit{M. von Renteln}, Collect. Math. 26, 115--126 (1975; Zbl 0315.46044)] and Tolokonnikov [\textit{V. A. Tolokonnikov}, J. Sov. Math. 27, 2549--2553 (1984; Zbl 0546.46046)]. Previous work is reviewed, and the paper proceeds to assemble a large number of partial results that have been worked out by the author, in part with P. Gorkin, over a 20 year period. The problem is connected to the trivial Gleason parts of \(H^\infty\), to the number of generators of an order \(N\) ideal, and the techniques include Carleson measures and Carleson-Newman Blaschke products (finite products of interpolating Blaschke products), and higher-order zero sets. A sample result (of 22 in all) is Theorem 4.3: Let \(b\) and \(c\) be two Carleson-Newman Blaschke products of order 2. Then the ideal \(J(b,c)\) is three-generated. Additionally, if \(b\) and \(c\) have no common zeros in \(\mathbb D\), then there exist interpolating Blaschke products \(B\), \(B^*\), \(C\) and \(C^*\) such thatNEWLINENEWLINE1. \(b=BB^*\), \(c=CC^*\),NEWLINENEWLINE 2. \(J(b,c)=I(BB^*,CC^*,BC)\).NEWLINENEWLINE In particular, we have \(J(BB^*,CC^*,BC)=I(BB^*,CC^*,BC)\).NEWLINENEWLINEThe paper also includes results about intersections of finitely-generated ideals, and a number of open problems are posed.
scientific article; zbMATH DE number 6084110
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